cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-5 of 5 results.

A175067 a(n) is the sum of perfect divisors of n, where a perfect divisor of n is a divisor d such that d^k = n for some k >= 1.

Original entry on oeis.org

1, 2, 3, 6, 5, 6, 7, 10, 12, 10, 11, 12, 13, 14, 15, 22, 17, 18, 19, 20, 21, 22, 23, 24, 30, 26, 30, 28, 29, 30, 31, 34, 33, 34, 35, 42, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 56, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 78, 65, 66, 67, 68, 69, 70, 71, 72
Offset: 1

Views

Author

Jaroslav Krizek, Jan 23 2010

Keywords

Comments

a(n) > n for perfect powers n = A001597(m) for m > 2.

Examples

			For n = 8: a(8) = 10; there are two perfect divisors of 8: 2 and 8; their sum is 10.

Links

Crossrefs

Cf. A175070, A007916, A175068, A253641, A052410.

Programs

  • Maple
    A175067:= proc(n) local a, d, k; if n=1 then return 1 end if; a:=0; for d in numtheory[divisors](n) minus {1} do for k do if d^k=n then a:= a+d end if; if n <= d^k then break; end if; end do; end do; a end proc:
seq(A175067(n), n=1..80); # Ridouane Oudra, Dec 12 2024
# second Maple program:
a:= n-> add(`if`(n=d^ilog[d](n), d, 0), d=numtheory[divisors](n)):
seq(a(n), n=1..72);  # Alois P. Heinz, Dec 12 2024
  • Mathematica
    Table[Plus @@ (n^(1/Divisors[GCD @@ FactorInteger[n][[All, 2]]])), {n, 72}] (* Ivan Neretin, May 13 2015 *)

Formula

a(n) = A175070(n) + n. [Jaroslav Krizek, Jan 24 2010]
From Ridouane Oudra, Dec 12 2024: (Start)
a(n) = n, for n in A007916.
a(n^m) = Sum_{d|m} n^d, for n in A007916 and m an integer >0.
More generally, for all integers n we have :
a(n) = Sum_{d|A253641(n)} n^(d/A253641(n)).
a(n) = Sum_{d|A253641(n)} A052410(n)^d. (End)

Extensions

Name edited by Michel Marcus, Jun 13 2018

A175084 Possible values for product of perfect divisors of n.

Original entry on oeis.org

1, 2, 3, 5, 6, 7, 8, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 26, 27, 28, 29, 30, 31, 33, 34, 35, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78
Offset: 1

Views

Author

Jaroslav Krizek, Jan 24 2010

Keywords

Comments

Perfect divisor of n is divisor d such that d^k = n for some k >= 1. All terms of this sequence occur only once. See A089723 (number of perfect divisors of n) and A175068 (product of perfect divisors of n).
Complement of A175085. - Jaroslav Krizek, Jan 30 2010

Crossrefs

Cf. A175087 (characteristic function).

Programs

  • Mathematica
    With[{nn = 78}, TakeWhile[#, # <= nn &] &@ Union@ Table[Apply[Times, Select[Divisors@ n, Or[# == 1, #^IntegerExponent[n, #] == n] &]], {n, nn}] ] (* Michael De Vlieger, Nov 21 2017 *)

A175087 Number of numbers whose product of perfect divisors is equal to n.

Original entry on oeis.org

1, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1
Offset: 1

Views

Author

Jaroslav Krizek, Jan 24 2010

Keywords

Comments

Perfect divisor of n is divisor d such that d^k = n for some k >= 1. See A175068 (product of perfect divisors of n), A175084 (possible values for product of perfect divisors of n) and A175085 (numbers m such that product of perfect divisors of x = m has no solution). a(n) = 0 or 1 for all n.
That is, this is the characteristic function of A175084. - Antti Karttunen, Nov 21 2017

Links

Crossrefs

Cf. A175068, A175084 (positions of ones), A175085 (of zeros).

Programs

  • Mathematica
    With[{nn = 105}, ReplacePart[ConstantArray[0, nn], Flatten@ Table[{i -> 1}, {i, TakeWhile[#, # <= nn &] &@ Union@ Table[Apply[Times, Select[Divisors@ n, Or[# == 1, #^IntegerExponent[n, #] == n] &]], {n, nn}]}] ] ] (* Michael De Vlieger, Nov 21 2017 *)
  • PARI
    A175068(n) = { my(m=1); fordiv(n,d,if((d>1)&&(d^valuation(n,d))==n,m*=d)); (m); };
A175087(n) = sum(i=1,n,A175068(i)==n); \\ Antti Karttunen, Nov 21 2017

Formula

a(n) = Sum_{k=1..n} [A175068(k)==n]. - Antti Karttunen, Nov 21 2017

Extensions

More terms from Antti Karttunen, Nov 21 2017

A175069 a(n) = product of perfect divisors of n / n.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 1, 2, 3, 1, 1, 1, 1, 1, 1, 8, 1, 1, 1, 1, 1, 1, 1, 1, 5, 1, 3, 1, 1, 1, 1, 2, 1, 1, 1, 6, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 7, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 64, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 27, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 10
Offset: 1

Views

Author

Jaroslav Krizek, Jan 23 2010

Keywords

Comments

A perfect divisor of n is a divisor d such that d^k = n for some k >= 1.

Links

Crossrefs

Cf. A175068.

Programs

  • Mathematica
    Table[Apply[Times, Select[Divisors@ n, Or[# == 1, #^IntegerExponent[n, #] == n] &]]/n, {n, 105}] (* Michael De Vlieger, Nov 21 2017 *)
  • PARI
    A175068(n) = { my(m=1); fordiv(n,d,if((1==d)||(d^valuation(n,d))==n,m*=d)); (m); };
A175069(n) = (A175068(n)/n); \\ Antti Karttunen, Nov 21 2017

Formula

a(n) = A175068(n) / n. a(n) > 1 for perfect powers n = A001597(m) for m > 2.

A175086 Perfect powers m such that product of perfect divisors of x = m has solution.

Original entry on oeis.org

1, 8, 16, 27, 64, 81, 125, 128, 216, 256, 343, 625, 729
Offset: 1

Views

Author

Jaroslav Krizek, Jan 24 2010

Keywords

Comments

Perfect divisor of n is divisor d such that d^k = n for some k >= 1. Subsequence of A001597 (perfect powers). See A175068 (product of perfect divisors of n) and A175086 (numbers m such that product of perfect divisors of x = m has no solution).
Showing 1-5 of 5 results.

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